arXiv subjects

Riccardo Pengo

Publications and source records attributed to Riccardo Pengo.

8 recordsLinked to original sources

arXiv2025

On Diophantine properties for values of Dedekind zeta functions

We study the Northcott and Bogomolov property for special values of Dedekind $ζ$-functions at real values $σ\in \mathbb{R}$. We prove, in particular, that the Bogomolov property is not satisfied when $σ\geq \frac{1}{2}$. If $σ> 1$, we produce certain families of number fields having arbitrarily large degrees, whose Dedekind $ζ$-functions $ζ_K(s)$ attain arbitrarily small values at $s = σ$. On the other hand, if $\frac{1}{2} \leq σ\leq 1$, we construct suitable families of quadratic number fields, employing either Soundararajan's resonance method, which works when $\frac{1}{2} \leq σ< 1$, or results on random Euler products by Granville and Soundararajan, and by Lamzouri, which work when $\frac{1}{2} < σ\leq 1$. We complete the study by proving that the Dedekind $ζ$ function together with the degree satisfies the Northcott property for every complex $s\in{\mathbb{C}}$ such that $\mathrm{Re}(s) <0$, generalizing previous work of Généreux and Lalín.

math.NT↗

arXiv2024

Irreducibility criteria for the preimages of a transverse variety under endomorphisms of products of elliptic curves

We provide two different proofs of an irreducibility criterion for the preimages of a transverse subvariety of a product of elliptic curves under a diagonal endomorphism of sufficiently large degree.For curves, we present an arithmetic proof of the aforementioned irreducibility result, which enlightens connections to methods used in the context of the Torsion Anomalous Conjecture. On the other hand, we generalize the result for higher dimensional varieties using a more geometric approach. Finally, we give some applications of these results. More precisely, we establish the irreducibility of some explicit families of polynomials, we provide new estimates for the normalized heights of certain intersections and images, and we give new lower bounds for the essential minima of images.

math.AG↗

arXiv2024

Spanning trees in $\mathbb{Z}$-covers of a finite graph and Mahler measures

Using the special value at $u=1$ of Artin-Ihara $L$-functions, we associate to every $\mathbb{Z}$-cover of a finite connected graph a polynomial which we call the \emph{Ihara polynomial}. We show that the number of spanning trees for the finite intermediate graphs of such a cover can be expressed in terms of the Pierce-Lehmer sequence associated to a factor of the Ihara polynomial. This allows us to express the asymptotic growth of the number of spanning trees in terms of the Mahler measure of this polynomial. Specializing to the situation where the base graph is a bouquet or the dumbbell graph gives us back previous results in the literature for circulant and $I$-graphs (including the generalized Petersen graphs). We also express the $p$-adic valuation of the number of spanning trees of the finite intermediate graphs in terms of the $p$-adic Mahler measure of the Ihara polynomial. When applied to a particular $\mathbb{Z}$-cover, our result gives us back Lengyel's calculation of the $p$-adic valuations of Fibonacci numbers.

math.NT↗

arXiv2023

On the Northcott property for special values of L-functions

We propose an investigation on the Northcott, Bogomolov and Lehmer properties for special values of L-functions. We first introduce an axiomatic approach to these three properties. We then focus on the Northcott property for special values of L-functions. In the case of L-functions of pure motives, we prove a Northcott property for special values located at the left of the critical strip, assuming that the L-functions in question satisfy some expected properties. Inside the critical strip, focusing on the Dedekind zeta function of number fields, we prove that such a property does not hold for the special value at one, but holds for the special value at zero, and we give a related quantitative estimate in this case.

math.NT↗

arXiv2022

Limits of Mahler measures in multiple variables

We prove that certain sequences of Laurent polynomials, obtained from a fixed Laurent polynomial P by monomial substitutions, give rise to sequences of Mahler measures which converge to the Mahler measure of P. This generalizes previous work of Boyd and Lawton, who considered univariate monomial substitutions. We provide moreover an explicit upper bound for the error term in this convergence, generalizing work of Dimitrov and Habegger, and a full asymptotic expansion for a family of 2-variable polynomials, whose Mahler measures were studied independently by the third author.

math.NT↗

arXiv2022

How big is the image of the Galois representations attached to CM elliptic curves?

Using an analogue of Serre's open image theorem for elliptic curves with complex multiplication, one can associate to each CM elliptic curve $E$ defined over a number field $F$ a natural number $\mathcal{I}(E/F)$ which describes how big the image of the Galois representation associated to $E$ is. We show how one can compute $\mathcal{I}(E/F)$, using a closed formula that we obtain from the classical theory of complex multiplication.

math.NT↗

arXiv2021

Entanglement in the family of division fields of elliptic curves with complex multiplication

For every elliptic curve $E$ which has complex multiplication (CM) and is defined over a number field $F$ containing the CM field $K$, we prove that the family of $p^{\infty}$-division fields of $E$, with $p \in \mathbb{N}$ prime, becomes linearly disjoint over $F$ after removing an explicit finite subfamily of fields. We then give a necessary condition for this finite subfamily to be entangled over $F$, which is always met when $F = K$. In this case, and under the further assumption that the elliptic curve $E$ is obtained as a base-change from $\mathbb{Q}$, we describe in detail the entanglement in the family of division fields of $E$.

math.NT↗

arXiv2020

Mahler's measure and elliptic curves with potential complex multiplication

Given an elliptic curve $E$ defined over $\mathbb{Q}$ which has potential complex multiplication by the ring of integers $\mathcal{O}_K$ of an imaginary quadratic field $K$ we construct a polynomial $P_E \in \mathbb{Z}[x,y]$ which is a planar model of $E$ and such that the Mahler measure $m(P_E) \in \mathbb{R}$ is related to the special value of the $L$-function $L(E,s)$ at $s = 2$.

math.NT↗